Element-finite Element Method Research Articles

How to achieve the fast computation for voxel-based irregular structures by few finite elements?

We propose machine learning (ML) based smart interpolation functions to enhance the finite element computation (named as smart-I finite element) for the heterogeneous structures (e.g., alloy material, concrete material and porous rock) that is composed of many microstructures known as voxel-based irregular shape (VI-shape). This element can directly describe the geometries with different VI-shapes and material properties without the complex meshing process by traditional finite element method (FEM). To achieve the smart interpolation in such element, a machine learning algorithm containing multiple parallel artificial neural networks (ANNs) and DeepONets assembled by the convolutional architectures is constructed for the fast prediction of interpolation functions. On this basis, the smart-I finite element is combined with classic FEM element to derive the global equilibrium equations and obtain the full-field variable distribution. Numerical tests with various material, geometrical as well as loading parameters prove that, the smart-I finite element enhanced FEM (SFEE-FEM) is able to conduct the accurate computation with a small number of elements, and thus greatly reduces the time cost. The smart-I finite element proposed by this work breaks through the limitation of traditional element shape with regular boundary (e.g., straight line), and may pave a new way to implement the FEM calculation on voxel-based geometries efficiently.

Finite Element Method approach 3-dimensional thermophysical model for YORP torque computation

We present a new thermophysical model suitable for simulating the thermal conditions of small celestial bodies, such as asteroids and comet nuclei, as well as local topographic features like rocks, boulders, and craters. The model employs a Finite Element Method (FEM) approach to solve the 3-dimensional heat conduction problem, explicitly accounting for heat conduction between neighboring FEM elements. This sets it apart from other thermophysical models that typically neglect lateral heat conduction. The model takes into account scattering sunlight, self-heating due to thermal radiation, and both horizon and cast shadows. We validated our model against an analytical solution for the transient temperature distribution of a sphere under convective boundary conditions. Additionally, we compared our results with earlier thermophysical modeling works and observational data of asteroid (101955) Bennu. Using the shape models of Bennu as well as scaled versions of (162173) Ryugu, (25143) Itokawa, and (6489) Golevka to match Bennu’s surface area, we investigated the effects of shape and lateral heat conduction on the Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) torque. Results indicate that contact binary and non-classified/irregular objects experience a stronger YORP torque compared to spheroidal objects. The lateral heat conduction can enhance or dampen the YORP torque. Further analysis is important to obtain more robust statistical conclusions.

Numerical analysis and experimental measurement on segmented ring transducer acoustic fields

In order to overcome the shortcomings of the two traditional methods of finite element method (FEM) and direct coupling of mechanical FEM and acoustic boundary element method (BEM) in the comprehensive simulation of acoustic characteristics of non-two-dimensional axisymmetrical transducers, the acoustic FEM and acoustic BEM coupling method is adopted in this study to model, calculate and analyze the three-dimensional (3D) model of the segmented ring transducer. Firstly, the basic theory of FEM, BEM and FEM-BEM coupling calculation of the transducer is introduced. Secondly, a 1/8 3D model of the segmented ring transducer is built to simulate the underwater electro-acoustic characteristics of the transducer. Finally, a transducer prototype is manufactured according to the numerical model, and the electro-acoustic performance of the transducer is measured in the anechoic tank. The numerical results of transmission voltage response (TVR), admittance and radiation directivity curves are in good agreement with the experimentally measured results, which indicate that the proposed method is correct and effective. Therefore, the method could be used to comprehensively calculate the acoustic characteristics of the segmented ring transducer. In addition, the method could be used not only to study the acoustic problems of other transducers or transducers arrays with complex structures that are non-two-dimensional axisymmetrical, but also to comprehensively analyze and predict the near-field and far-field acoustic characteristics of any transducers including two-dimensional (2D) axisymmetrical transducers.

Numerical simulation of impact and entrainment behaviors of debris flow by using SPH–DEM–FEM coupling method

Abstract Increasing rain levels can easily destabilize and destroy particulate matter in mountainous areas, which can cause natural disasters, such as debris flow and landslides. Constitutive equations and numerical simulation are the theoretical bases for understanding the behavior of these disasters. Thus, this study aimed to investigate the impact of the debris flow and its entrainment behavior on gully bed sediments. We adopted a coupled analysis method based on elastic–plastic constitutive equations by considering the elasto-plasticity of slurry and the elastic characteristics of debris materials. The coupled method consisted of smooth particle hydrodynamic (SPH), discrete element method (DEM), and finite element method (FEM) (SPH–DEM–FEM). SPH particles represented fluid, DEM particles denoted solid immersed in fluid, and FEM elements represented the terrain and structures. The coupling analysis model was used to simulate the coupling contact of solid, liquid, and structures and to describe the entrainment behavior between solid and liquid phases. The model feasibility was verified by comparing the basic simulation results with experimental values of the dam break model and the rotating cylindrical tank model. The coupled model was then combined with the data management and modeling of geographic information system to simulate the 2010 Yohutagawa debris flow event. Finally, we explored the influence of debris shape-related parameters on the debris flow erosion entrainment process.

Thermal-Mechanical Coupling Model Based on the Hybrid Finite Element Method for Solving Bipolar the Plate Deformation of Hydrogen Fuel Cells

New energy is the focus of attention all over the world, and research into new energy can inject new vitality into the industrial system. Hydrogen fuel cells are not only environmentally friendly, but also rich in reserves that can be used as a strategic resource for the entire country. The difficulty lies in the safe design of application equipment and the batch generation and storage of hydrogen. In addition, fuel cells have the disadvantage of a slow start-up. Based on the above problems, this paper proposes a hybrid-element method to solve the thermal-mechanical coupling model of fuel cell plate, which can effectively solve the thermal stress change, temperature field distribution and displacement change of the battery plate when working. Firstly, the hybrid-element algorithm is given for 2D plate deformation. Then, the deformation application of a 3D fuel cell plate is given. The 2D numerical results show that the hybrid finite element method (FEM) is more flexible for realizing the flexible combination of sub-mesh and finite element basis functions, and has a better mesh quality compared to the traditional constant strain triangular element (CST) adaptive FEM and quadrilateral isoparametric element (Q4) adaptive FEM. This method achieves a balance between numerical accuracy and solving efficiency for the multi-porous elastic plate. In addition, a deformation control formula is given which can display the displacement deformation and stress merge to same graph, since it is convenient to quickly compare the regions where the displacement and stress extremum appear. In short, the hybrid finite element method proposed in this paper has good mesh evaluation results, and when the number of discrete elements is equivalent, the hybrid element converges faster and the solution efficiency is higher. This paper also provides a good numerical theory and simulation reference for industrial mechanics and new energy applications.

A simple particle–spring method for capturing the continuous–discontinuous processes of brittle materials

Comparing to the continuum-based method such as the finite element method (FEM), particle–spring method can relatively easily simulate the cracking processes of brittle materials such as rocks by deactivating or adjusting springs/contacts. However, it is still challenging for building the equivalence between particle–spring system and the original continuous media, then capturing the whole continuous–discontinuous processes. In this work, a simple particle–spring system is proposed in which different types of springs are introduced including normal, tangential, pure shearing and Poisson’s springs. The stiffness of the springs are obtained by bridging the proposed model and triangular FEM element. Moreover, novel mixed type failure criteria for the breakage of the springs are also provided. By numerical studies the reliability and robustness of the model are proven which gives comparable results to the conventional FEM model. With the new model, progressive failure processes of rock-like material can be properly captured.

A Novel Solution for Seepage Problems Implemented in the Abaqus UEL Based on the Polygonal Scaled Boundary Finite Element Method

The scaled boundary finite element method (SBFEM) is a semianalytical computational scheme based on the characteristics of the finite element method (FEM) and boundary element method that combines their respective advantages. In this paper, the SBFEM and polygonal mesh technique are integrated into a new approach to solve steady-state and transient seepage problems. The proposed method is implemented in Abaqus employing a user-defined element (UEL). A detailed implementation of the procedure is presented in which the UEL element is defined, the internal variables RHS and AMATRX are updated, and the stiffness/mass matrix is solved using eigenvalue decomposition. Several benchmark problems are solved to verify the proposed implementation. The results show that the polygonal element of the polygonal SBFEM (PSBFEM) is more accurate than the standard FEM element of the same element size. For transient problems, the results for the PSBFEM and FEM are in excellent agreement. Hence, the proposed method is robust and accurate for solving steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from GitHub.

Estimation of Bearing Capacity of Strip Footing Rested on Bilayered Soil Profile Using FEM‐AI‐Coupled Techniques

In this research work, finite element method (FEM) and 2D Plaxis were employed to generate numerical values for bilayered soils bearing a strip footing of width B and depths, h and H, in order to predict the ultimate bearing capacity (UBC) of the strip footing underlain by layered soil profile. Several research works have tried to solve bearing capacity problems using limit equilibrium (LE) techniques. But, the LE techniques have limitations in terms of soil properties and profile arrangement. The need, however, for using constitutive models or numerical methods powered by FEM and discrete element method (DEM) has been on the rise due to the versatility and robustness of these techniques to accommodate erratic soil behaviors. Multiple numerical data were generated for the case under study and artificial intelligence (AI)‐based techniques; generalized reduced gradient (GRG), genetic programming (GP), artificial neural network (ANN), and evolutionary polynomial regression (EPR) were used to predict the UBC. In order to conduct the parametric analysis to investigate the effect of different soil layers, footing width and overburden pressure at foundation level on the ultimate bearing capacity sets of finite element models were prepared. This was executed using the following soil properties: soil type of top layer (from S1 to S6), soil type of bottom layer (from S1 to S6), strip footing width (B) (from 1.0 to 5.0 m), thickness of the top layer (h) (from 0.5 B to 1.0 B), overburden pressure (σ′v) (from 1.0 m to 3.0 m multiplied by the γ′t), and the parameters combination of each finite model in the set is randomly selected. The results of the FEM/Plaxis parametric study produced the 2D‐model, deformed shape, stress distribution, and plastic point (failure point) models. The loading produced appreciable deformation on both x‐ and y‐axes of the soil profile with the y‐axis showing a scattered failure configuration. The AI‐based prediction produced UBC equations which performed at over 90% accuracy with ANN (99.9%; 6.3%) outperforming other techniques followed by GRG, GP, and EPR.

Effects of non-axisymmetric internal structures on vibro-acoustic characteristics of a submerged cylindrical shell using wavenumber analysis

Non-axisymmetric structures (foundations, equipment etc.) have been extensively arranged in axisymmetric shells and proved to change the mechanism of vibration and sound radiation of the shell. To study the effect of non-axisymmetric internal structures, a hybrid analytic-numerical method is developed for vibro-acoustic characteristics analyses of a submerged cylindrical shell. The wave based method (WBM) and finite element method (FEM) are respectively employed to model the shell and internal structures. A precise coupling is carried out to assemble the shell and internal structures, which fully considers all six degree of freedoms (DOFs) at junctions between the two substructures. The acoustic pressure is described by the Helmholtz integral equation. According to the axisymmetric property of the shell, the surface integral is reduced into the curve integral. Subsequently, the final vibro-acoustic model is obtained by introducing the acoustic pressure into the dynamic equation of the coupled structure. The high accuracy of the present method is firstly demonstrated by FEM and boundary element method (BEM). Then by comparing vibro-acoustic responses of the shell with and without internal structures in wave domain, the effects of non-axisymmetric internal structures on vibro-acoustic characteristics of the shell are discussed.

A weak-scatterer potential flow theory-based model for the hydroelastic analysis of offshore wind turbine substructures

This paper examines the hydroelastic response of a monopile structure, supporting an offshore wind turbine. A new numerical simulation tool is presented, coupling a nonlinear potential flow solver to a structural model based on modal superposition. The hydrodynamic solver is based on the Weak-Scatterer (WS) approach and assumes small perturbations of the incident flow. A finite element method (FEM) solver is used to compute the modal parameters, which are then superposed to compute the dynamics of the system in time domain. Small deformations are assumed in the coupling. The two theories are tightly coupled in time domain. The theory of the coupling is fully described in the paper. The new coupled solver WS_CN-FEM is then applied to the case of a large diameter monopile. Results are compared to simulations using the Morison equation to compute the hydrodynamic loads and a beam element FEM model to compute the response of the structure and to experimental measurements made on a monopile-based offshore wind turbine model. The physical model has Froude-scaled geometry and natural frequencies, which allows an accurate validation of the hydroelastic numerical models including realistic flexible modes of the structure and wave–structure interaction. The results of WS_CN-FEM show a good agreement with the experimental measurements, and in particular on the first and second mudline bending moments’ harmonics in a series of regular waves of various wave steepness.

Analysis of Load Noise Components in Small Core-Form Transformers

We propose load noise analysis components for small core-form transformers (1 MVA–20 MVA) using computational fluid-structure coupled calculations that consider the electromagnetic force, transformer structure and fluid (oil). The proposed technique has two beneficial aspects: coil modeling using shell elements of the finite element method (FEM), and acoustic noise analysis using a combination of boundary element method transmission path simulation and FEM structural simulation. The shell element model, which includes structural bending information, is less computationally intensive for simulations than solid models because it uses larger element sizes. Lighter simulations enable easier redesign for noise reduction. Additionally, division of the simulation process into individual structural vibrations, transmission paths, and radiation processes makes it easier to find a remodeling point without performing a complete simulation. We illustrate the importance of avoiding coincidence of the natural frequencies of the coils and the electromagnetic force and present a practical noise reduction study. This technique can be implemented using a light simulation that was proposed to confirm the remodeling effects. Using the proposed analysis method along with a ray tracing method, we can construct a frontloading design simulation tool to determine the transformer specifications and the substation's geographical features.

High-fidelity modelling of thermal stress for additive manufacturing by linking thermal-fluid and mechanical models

The prediction of thermal stress and distortion is a prerequisite for high-quality additive manufacturing (AM). The widely applied thermo-mechanical model using the finite element method (FEM) leaves much to be improved due to their oversimplifications on material deposition, molten pool flow, etc. In this study, a high-fidelity modelling approach by linking the thermal-fluid (computational fluid dynamics, CFD) and mechanical models (named as CFD-FEM model) is developed to predict the thermal stress for AM taking into account the influences of thermal-fluid flow. Profiting from the precise temperature profiles and melt track geometry extracted from the thermal-fluid model as well as the remarkable flexibility of the quiet element method of FEM, this work aims at simulating the thermal stress distribution by involving physical changes in the AM process, e.g., melting and solidification of powder particles, molten pool evolution and inter-track inter-layer re-melting. Unlike the conventional thermo-mechanical analysis, in this approach, thermal stress calculation is purely based on a mechanical model where the thermal loads are applied by using a linear interpolation function to spatially and temporally map the temperature values from the thermal-fluid model's cell centres into the FEM element nodes. With the proposed approach, the thermal stress evolution in the AM process of single track, multiple tracks and multiple layers are simulated, where the rough surfaces and internal voids can be well incorporated. Moreover, a conventional thermo-mechanical simulation of two tracks with predefined track geometry is conducted for cross comparison. Finally, the simulated thermal stress distribution can rationally explain the crack distribution observed in the experiments.

Wrinkling Analysis of Pre-Stressed Rectangular Membranes Using Element-Free Galerkin Method

In this study, element-free Galerkin method (EFGM), a meshless method, is proposed for wrinkling analysis of pre-stressed rectangular membranes. The mathematical model for studying wrinkling of pre-stressed membranes is derived by considering the bending stiffness, though it is negligible. Moving least-square approximation for deflection is constructed by considering three degrees of freedom per node. Essential boundary conditions are imposed using scaled transformation matrix method. Initially, compression-induced buckling of a homogeneous thin plate without pre-stress is solved to validate the method and then a pre-stressed homogeneous membrane is analyzed for both compression-induced and shear-induced wrinkling. Capabilities of the proposed method for membrane analysis are compared with that of the finite element method (FEM). Comparative study on wrinkling analysis using EFGM and different FEM element types in a commercial FEM package shows that in lower modes both methods show satisfying consistency in eigenvalues with respect to the total of number of nodes, while at higher modes EFGM shows better consistency than FEM. Further, the study is extended to wrinkling of nonhomogeneous membranes subjected to linearly-varying in-plane load. The results obtained from EFGM analysis is compared and found to be matching well with those available in the literature.

A novel Lagrangian finite element method with adaptive element-particle conversion ability for incompressible flows with free surfaces

When modeling fluid flows with the finite element method (FEM), a normal way is to implement the simulation in an Eulerian frame, as it is difficult for the Lagrangian FEM to deal with fluid flows with extremely large deformations, especially for incompressible flows with violent fluid-structure interactions along with changing free surfaces. In this paper, we developed a novel Lagrangian FEM for modeling incompressible flows with violently changing free surfaces. The Lagrangian FEM is an edge-based smoothed FEM (ES-FEM) which has special advantages over the conventional FEM due to the gradient smoothing technique. An artificial equation of state is used in the ES-FEM to treat the incompressible flow as weakly compressible. In order to simulate large fluid deformation, an adaptive algorithm is employed to convert the FEM elements in areas with large fluid deformation into particles in the smoothed particle hydrodynamics (SPH) as the SPH method is well suited for modeling fluid deformation and free surfaces. Some state-of-art algorithms including the kernel gradient correction and particle shift technique are integrated into the SPH method to ensure the computational accuracy of the fluid modeling. After such treatment, the Lagrangian ES-FEM is used for modeling fluid flows with small deformations while SPH is employed for areas with large fluid deformations where ES-FEM elements are automatically converted to SPH particles. Four numerical examples including still water, lid driven cavity flow, dam break and water entry are modeled using the present Lagrangian FEM. The numerical studies demonstrate that the present Lagrangian FEM with adaptive element-particle conversion ability is as accurate as SPH in modeling incompressible flows with violently changing free surfaces while it is much more efficient than SPH.

Shape optimization of conical hoppers to increase mass discharging rate

Mass discharging rate (MDR) is a critical aspect of hopper’s performance in bulk solids handling. A shape optimization method is established in this study to increase the MDR of cohesionless granular materials from hoppers. This method is based on a continuum model of granular matter and the Eulerian Finite Element Method (FEM) which can efficiently simulate the discharging process and predict the MDR. In this work with the focus on conical hoppers, the widths of silo and hopper outlet as well as the vertical height of hopper are fixed. The meridian of the hopper, however, evolves from a straight line to some optimal curve, guided by a combined genetic algorithm (GA) and gradient descent method (GDM). Cubic spline function is employed to parametrize the hopper shape. The effectiveness of the shape optimization is examined by comparing the MDRs of the optimal hopper and conventional conical hopper, obtained by both FEM and discrete element method (DEM) respectively. It is shown that this shape optimization method can automatically search the optimal shape of the hopper in a given range of constraints, and increase the MDR substantially. In a typical hopper with an initial half angle of 45°, the MDR is increased by over 130% after the shape optimization. Notably, the optimal shape depends mainly on the geometrical factors, i.e. the allowed width and height for the hopper, whilst insensitive to the material properties, which favors its general use for different particles. Such curved hoppers are particularly useful for increasing the discharge rate of hoppers ranging from 30° to 50°, which, facilitated with advanced manufacturing technology, will find wide potential applications in bulk solids handling.

Monolithic coupling of the implicit material point method with the finite element method

A monolithic coupling between the material point method (MPM) and the finite element method (FEM) is presented. The MPM formulation described is implicit, and the exchange of information between particles and background grid is minimized. The reduced information transfer from the particles to the grid improves the stability of the method. Once the residual is assembled, the system matrix is obtained by means of automatic differentiation. In such a way, no explicit computation is required and the implementation is considerably simplified. When MPM is coupled with FEM, the MPM background grid is attached to the FEM body and the coupling is monolithic. With this strategy, no MPM particle can penetrate a FEM element, and the need for computationally expensive contact search algorithms used by existing coupling procedures is eliminated. The coupled system can be assembled with a single assembly procedure carried out element by element in a FEM fashion. Numerical results are reported to display the performances and advantages of the methods here discussed.

A Generalized Parallel Transmission Line Iteration for Finite Element Analysis of Permanent Magnet Axisymmetrical Actuator

At present, finite element method (FEM) is still the mainstream tool in a range of complex science and engineering applications, including the analysis of structures, heat, fluid, and electromagnetics. However, FEM becomes computationally very intensive when the number of physics equations grows and the mesh is refined. In this paper, we investigated the parallel computing techniques in the nonlinear magnetostatic field solution of an axisymmetrical actuator with permanent magnet by proposing a generalized black-box transmission line method (BB-TLM). A novel black-box circuit model was built to represent complex FEM element data. By means of the transmission line model, each element is isolated and then a series of parallel procedures is considered during the solution stages. The magnetic field distribution and magnetic force of the studied actuator are calculated by the proposed method and the simulation results are compared with COMSOL. Compared with the conventional N-R method, the generalized BB-TLM algorithm greatly reduces the computation time.

Numerical analysis of drying process of soils using finite volume method

The accurate prediction of the moisture content in soil is important for pavement engineering. The MEPDG uses the Enhanced Integrated Climate Model (EICM) to consider the effects of environment on the moisture contents of unbound and subgrade soil using models related to unsaturated hydraulic conductivity. These models are mostly empirical and not applicable to relatively dry conditions. This is because at relatively dry condition, the moisture in the pore structure of the soil is not inter-connected. Therefore, the moisture diffusion in porous material controls the moisture migration. When the diffusion coefficient is a nonlinear function of pore relative humidity (RH), there is no closed-form solution of the constitutive differential equation of the moisture diffusion. This study used finite-volume method (FVM) and finite-element method (FEM) for the numerical simulation of the moisture diffusion in soils. The FVM, which is similar to the FEM, uses small and finite-sized elements for simulation, but is based on the law of conservation. Therefore, FVM will be more suitable for flux conservation problems such as moisture diffusion. The FVM results were verified with laboratory experiments and compared with FEM results. The results indicate the applicability of using FVM in the simulation of the moisture migration in soils.

Smoothed finite element method for time dependent analysis of quantum resonance devices

In this paper, a new finite element method (FEM) is introduced to study the time-dependent wave nature of the electron in quantum resonance devices. Unlike the well-known FEM, the new method smooths the wave function derivatives over the edges. In this sense, the new method is termed “smoothed FEM” where an “inter-element” matrix is formed to smooth the derivatives over the edges. For the electron’s wave function propagation in time, the presented method exploits the time domain beam propagation method (TD-BPM). Based only on first order elements, our suggested SFETD-BPM has high accuracy levels comparable to second-order conventional FEM elements; thanks to the element smoothing. The proposed method numerical performance is tested through the analysis of a quantum resonance cavity and a quantum resonant tunneling device. It is clearly demonstrated that the presented method is not only accurate but also more time efficient than the conventional FEM approach.

An efficient classification-reconstruction method for 3D EIT imaging

Abstract Electrical Impedance Tomography (EIT) is a radiation-free imaging technique. The reconstructed EIT image represents the conductivity distribution changes of the human body. EIT images are reconstructed based on the Finite Element Method (FEM). Currently, the 3D EIT reconstruction is desired. However, because of the huge amount of parameters associated with the FEM model, the 3D EIT reconstructions are neither stable nor computationally efficient. In this paper, we present a classification-reconstruction method for EIT imaging. In the classification step, based on a fast initial linear reconstruction, the FEM elements are grouped into several clusters. The reconstructions are based on these clusters by forcing the conductivity distribution the same within each cluster. The proposed framework is efficient in computation and potentially fit for 3D EIT reconstruction.